Problem 13 Graph the function. $$h(x)=(x-... [FREE SOLUTION]

Chapter 4: Problem 13

Graph the function. $$h(x)=(x-3)\left(x^2+x+1\right)$$

Short Answer

Expert verified

This is a fairly complicated function to graph by hand, and it may be easier to use graphing software. However, by identifying zeros and extrema, a rough sketch can be made. The graph is a upward-opening curve with x-intercepts at the solutions of $x^3 - 2x^2-3x +3 = 0$ and turning points at the solutions of $3x^2 - 4x - 3 = 0$.

Step by step solution

Expand the function

Expand the given function $ h(x)=(x-3)\left(x^2+x+1\right) $ by multiplying the terms inside the parentheses by $ x-3 $. This yields $ h(x) = x^3 - 2x^2-3x +3 $

Identify the zeros

Set the function equal to zero and solve for $x$. This means solving the equation $x^3 - 2x^2-3x +3 = 0$. The solutions to this equation are the x-intercepts of the function on a graph.

Identify the extrema

Take the first derivative of the function and set equal to zero. This will yield the x-values where the function has a local maximum or minimum (extrema). The first derivative is $h'(x) = 3x^2 - 4x - 3 $. Solving $3x^2 - 4x - 3 = 0$ will provide these x-values.

Graph the function

Using the zeros and extrema values, sketch the graph of the function. Start by marking the x-intercepts, then draw the curve upwards or downwards at extrema points. Make sure the function aligns with these critical points.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Functions

Polynomial functions are a type of mathematical expression that involve sums of powers of a variable, typically written in terms of coefficients multiplied by powers of a variable, like x. For example, in the function $h(x) = (x-3)(x^2+x+1)$, when we expand it, it becomes $x^3 - 2x^2 - 3x + 3$. This expression is a third-degree polynomial because it has the term $x^3$, which is the highest power of x in the expression.

Polynomial functions are characterized by their degree, which tells you the highest power of x and usually indicates the general shape and number of turning points on a graph.
They can have various shapes and complexities, but they are always smooth and continuous.
The coefficients of the powers of x influence the spread, size, and orientation of the graph.

Understanding polynomial functions starts with recognizing these expressions and being able to expand and simplify them.

Zeros of a Function

The zeros of a function are the x-values where the function equals zero. For the polynomial $h(x) = x^3 - 2x^2 - 3x + 3$, finding the zeros involves setting the equation to zero: $x^3 - 2x^2 - 3x + 3 = 0$, and solving for x.

The zeros are also known as roots or x-intercepts and they are crucial for understanding the behavior of the function graphically.
They represent the points where the graph of the function crosses or touches the x-axis.
Zeros can be found using various methods, such as factoring, using the quadratic formula for simpler polynomials, or employing numerical techniques for more complex polynomials.

Zeros not only help in sketching the graph but also provide insights into the factorization of the polynomial, as each zero represents a factor of the polynomial function.

First Derivative

The first derivative of a function provides information about the rate of change of the function and is essential in understanding the slope of the tangent line at any given point. For our polynomial $h(x)$, the first derivative is found using basic calculus rules, yielding $h'(x) = 3x^2 - 4x - 3$.

The first derivative helps identify whether the function is increasing or decreasing at a particular point. If $h'(x) > 0$, the function is increasing; if $h'(x) < 0$, it is decreasing.
Critical points where $h'(x) = 0$ are important because they mark where the function might have a local maxima or minima.
Understanding the behavior of a function鈥檚 derivative is key for sketching the function accurately and determining how the curve fluctuates.

The first derivative is a powerful tool in calculus for decoding the behavior of polynomial functions and their graphs.

Extrema of a Function

Extrema refer to the maximum and minimum values that a function can reach, either locally (within a certain range) or globally (overall). To find the extrema of our polynomial $h(x) = x^3 - 2x^2 - 3x + 3$, we set its first derivative $h'(x) = 3x^2 - 4x - 3$ equal to zero and solve for x. This tells us the x-values where the function has potential local maxima or minima.

At these points, the graph of the function changes direction, which is why understanding these points is fundamental for graphing polynomials accurately.
Local maxima are the highest points in their vicinity, while local minima are the lowest.
Determining these points requires solving a quadratic equation that emerges from setting the derivative to zero.

These extremum points are critical for sketching the graph as they highlight turning points in the polynomial, offering a full picture of its behavior across different intervals.

91影视

Graph the function. $$h(x)=(x-3)\left(x^2+x+1\right)$$

Short Answer

Step by step solution

Expand the function

Identify the zeros

Identify the extrema

Graph the function

Key Concepts

Polynomial Functions

Zeros of a Function

First Derivative

Extrema of a Function

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